with \(1(\cdot )\)  being the indicator function. Following [1]}, the expression in (REF ) can be rearranged so that the conditional ES can be expressed in terms of the conditional AL scale parameter \(\delta _{tj}\) . Specifically, blackrecalling that each marginal of the MAL distribution has a univariate AL density with conditional scale parameter equal to \(\delta _{tj}={\mathbb {E}[(Y_{tj} - {\cal Q}_{Y_{tj}})(\tau _j - 1_{(Y_{tj} < {\cal Q}_{Y_{tj}})})]}\) , then equation (REF ) reduces to:
\(ES_{tj} =\mathbb {E}[Y_{tj}]-\frac{\delta _{tj}}{\tau _j}, \quad \quad t=1,2,...,T, \quad j=1,2,...,p\) 
